Math Gold Medalist

Lor

2023 AMC 10A

Problem 23

If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$, then $a$ and $b$ are said to be $\textit{complementary}$ divisors of $c$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. What is the sum of the digits of $N$?

$\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

Important Identities

Factorization

Prime Factorization

a(a+20)=b(b+23)

(2b+23)^2 – (2a+20)^2 = 129

2023 AMC 10B Problem 14

2018 BMO Round 1 Problem 3 (British Mathematical Olympiad)

   Solution